Experimental investigation of an interior search method within a simple framework

A steepest gradient method for solving Linear Programming (LP) problems, followed by a procedure for purifying a non-basic solution to an improved extreme point solution have been embedded within an otherwise simplex based optimiser. The algorithm is designed to be hybrid in nature and exploits many aspects of sparse matrix and revised simplex technology. The interior search step terminates at a boundary point which is usually non-basic. This is then followed by a series of minor pivotal steps which lead to a basic feasible solution with a superior objective function value. It is concluded that the procedures discussed in this paper are likely to have three possible applications, which are (i) improving a non-basic feasible solution to a superior extreme point solution, (iii) an improved starting point for the revised simplex method, and (iii) an efficient implementation of the multiple price strategy of the revised simplex method

Objective acceleration for unconstrained optimization

Acceleration schemes can dramatically improve existing optimization procedures. In most of the work on these schemes, such as nonlinear Generalized Minimal Residual (N-GMRES), acceleration is based on minimizing the $\ell_2$ norm of some target on subspaces of $\mathbb{R}^n$. There are many numerical examples that show how accelerating general purpose and domain-specific optimizers with N-GMRES results in large improvements. We propose a natural modification to N-GMRES, which significantly improves the performance in a testing environment originally used to advocate N-GMRES. Our proposed approach, which we refer to as O-ACCEL (Objective Acceleration), is novel in that it minimizes an approximation to the \emph{objective function} on subspaces of $\mathbb{R}^n$. We prove that O-ACCEL reduces to the Full Orthogonalization Method for linear systems when the objective is quadratic, which differentiates our proposed approach from existing acceleration methods. Comparisons with L-BFGS and N-CG indicate the competitiveness of O-ACCEL. As it can be combined with domain-specific optimizers, it may also be beneficial in areas where L-BFGS or N-CG are not suitable.Comment: 18 pages, 6 figures, 5 table

Coupling problem in thermal systems simulations

Building energy simulation is playing a key role in building design in order to reduce the energy consumption and, consequently, the CO2 emissions. An object-oriented tool called NEST is used to simulate all the phenomena that appear in a building. In the case of energy and momentum conservation and species transport, the current solver behaves well, but in the case of mass conservation it takes a lot of time to reach a solution. For this reason, in this work, instead of solving the continuity equations explicitly, an implicit method based on the Trust Region algorithm is proposed. Previously, a study of the properties of the model used by NEST-Building software has been done in order to simplify the requirements of the solver. For a building with only 9 rooms the new solver is a thousand times faster than the current method

MAGMA: Multi-level accelerated gradient mirror descent algorithm for large-scale convex composite minimization

Composite convex optimization models arise in several applications, and are especially prevalent in inverse problems with a sparsity inducing norm and in general convex optimization with simple constraints. The most widely used algorithms for convex composite models are accelerated first order methods, however they can take a large number of iterations to compute an acceptable solution for large-scale problems. In this paper we propose to speed up first order methods by taking advantage of the structure present in many applications and in image processing in particular. Our method is based on multi-level optimization methods and exploits the fact that many applications that give rise to large scale models can be modelled using varying degrees of fidelity. We use Nesterov's acceleration techniques together with the multi-level approach to achieve $\mathcal{O}(1/\sqrt{\epsilon})$ convergence rate, where $\epsilon$ denotes the desired accuracy. The proposed method has a better convergence rate than any other existing multi-level method for convex problems, and in addition has the same rate as accelerated methods, which is known to be optimal for first-order methods. Moreover, as our numerical experiments show, on large-scale face recognition problems our algorithm is several times faster than the state of the art